Absolutely he is allowed to bring up past crimes because it is a testament as he going to argue to your pattern and practice and character. It doesn't matter whether the public defender or judge deem it inappropriate, the fact remains your convictions can be raised in court. 

Now, in terms of pleading no contest (called nolo contendre), it is used to get rid of the possibility of possible civil actions or other such events.  In terms of pleading your case, if you changed it to not guilty, you now get the opportunity to present your case in court, bearing in mind you can get less or more of any prior plea arrangement with the prosecutor. 

If this is your first DUI and your prior convictions weren't serious, it will be your public defender's responsibility or duty to arrange a plea agreement, dropping the charge or pleading your case in court.